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Tame dynamics and robust transitivity chain-recurrence classes versus homoclinic classes


Authors: C. Bonatti, S. Crovisier, N. Gourmelon and R. Potrie
Journal: Trans. Amer. Math. Soc. 366 (2014), 4849-4871
MSC (2010): Primary 37C20, 37D30, 37C29, 37G25
Published electronically: May 8, 2014
MathSciNet review: 3217702
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Abstract: One main task of smooth dynamical systems consists in finding a good decomposition into elementary pieces of the dynamics. This paper contributes to the study of chain-recurrence classes. It is known that $ C^1$-generically, each chain-recurrence class containing a periodic orbit is equal to the homoclinic class of this orbit. Our result implies that in general this property is fragile.

We build a $ C^1$-open set $ \mathcal {U}$ of tame diffeomorphisms (their dynamics only splits into finitely many chain-recurrence classes) such that for any diffeomorphism in a $ C^\infty $-dense subset of $ \mathcal {U}$, one of the chain-recurrence classes is not transitive (and has an isolated point). Moreover, these dynamics are obtained among partially hyperbolic systems with one-dimensional center.


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Additional Information

C. Bonatti
Affiliation: CNRS - IMB, UMR 5584, Université de Bourgogne, 21004 Dijon, France
Email: bonatti@u-bourgogne.fr

S. Crovisier
Affiliation: CNRS - LMO, UMR 8628. Université Paris-Sud 11, 91405 Orsay, France
Email: sylvain.crovisier@math.u-psud.fr

N. Gourmelon
Affiliation: IMB, UMR 5251, Université Bordeaux 1, 33405 Talence, France
Email: Nicolas.Gourmelon@math.u-bordeaux1.fr

R. Potrie
Affiliation: CMAT, Facultad de Ciencias, Universidad de la República, Uruguay
Email: rpotrie@cmat.edu.uy

DOI: https://doi.org/10.1090/S0002-9947-2014-06261-2
Received by editor(s): January 10, 2012
Received by editor(s) in revised form: December 10, 2012
Published electronically: May 8, 2014
Additional Notes: The authors were partially supported by the ANR project DynNonHyp BLAN08-2 313375.
Article copyright: © Copyright 2014 American Mathematical Society